We discuss the integrability properties of the Boussinesq equations in the language of geometrical quantities defined on an appropriately chosen coset manifold connected with the W 3 algebra of Zamolodchikov. We provide a geometrical interpretation to the commuting conserved quantities, Lax-pair formulation, zero-curvature representation, Miura maps, etc. in the framework of nonlinear realizati...

A derivation of the Lax pair for the (1 + 1)-dimensional non-linear σ-model is described. Its main benefit is to have a clearer physical origin and to allow the study of a generalization to higher dimensions. PACS: 11.10.Kk, 11.30.-j , 11.30.Ly

A q-discrete version of the two-dimensional Toda molecule equation is proposed through the direct method. Its solution, Bäcklund transformation and Lax pair are discussed. The reduction to the q-discrete cylindrical Toda molecule equation is also discussed. 1 On leave from Department of Applied Mathematics, Faculty of Engineering, Hiroshima University. 1

The Calogero equation is used to illustrate the following two aspects of the Painlevé analysis of PDEs: (i) the singular expansions of solutions around characteristic hypersurfaces are neither single-valued functions of independent variables nor single-valued functionals of data; (ii) the truncated singular expansions not necessarily lead to the simplest, elementary, Bäcklund autotransformation...

We discuss the integrability properties of the Boussinesq equations in the language of geometrical quantities defined on an appropriately chosen coset manifold connected with the W 3 algebra of Zamolodchikov. We provide a geometrical interpretation to the commuting conserved quantities, Lax-pair formulation, zero-curvature representation, Miura maps, etc. in the framework of nonlinear realizati...

We study the complex Sharma–Tasso–Olver equation using Riemann–Hilbert approach. The associated problem for this integrable can be naturally constructed by considering spectral of Lax pair. Subsequently, in case that is irregular, N -soliton solutions deduced. In addition, three-dimensional graphic soliton and wave propagation image are graphically depicted further discussed.

We consider Fredholm determinants of matrix Hankel operators associated to versions the n-th Airy functions. Using theory integrable operators, we relate them a fully noncommutative Painlevé II hierarchy, defined through matrix-valued version Lenard operators. In particular, Riemann-Hilbert techniques used study these allows find Lax pair for each member hierarchy. Finally, coefficients matrice...

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peakons. The 3CH model is proven integrable in the sense of Lax pair, Hamiltonian structure, conservation laws. We show that admits peaked soliton (peakon) multi-peakon solutions. Additionally, reductions are investigated so new perturbed CH equation generated to possess peakon